| L(s) = 1 | + 4.22·2-s + 9.89·4-s + 5·5-s + 8.00·8-s + 21.1·10-s + 65.5·11-s − 7.45·13-s − 45.2·16-s + 73.6·17-s + 140.·19-s + 49.4·20-s + 277.·22-s − 58.4·23-s + 25·25-s − 31.5·26-s + 22.2·29-s − 283.·31-s − 255.·32-s + 311.·34-s − 264.·37-s + 592.·38-s + 40.0·40-s + 392.·41-s + 231.·43-s + 648.·44-s − 247.·46-s + 90.0·47-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.23·4-s + 0.447·5-s + 0.353·8-s + 0.668·10-s + 1.79·11-s − 0.159·13-s − 0.707·16-s + 1.05·17-s + 1.69·19-s + 0.553·20-s + 2.68·22-s − 0.529·23-s + 0.200·25-s − 0.237·26-s + 0.142·29-s − 1.64·31-s − 1.41·32-s + 1.57·34-s − 1.17·37-s + 2.52·38-s + 0.158·40-s + 1.49·41-s + 0.822·43-s + 2.22·44-s − 0.792·46-s + 0.279·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.012582865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.012582865\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.22T + 8T^{2} \) |
| 11 | \( 1 - 65.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 7.45T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 140.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 58.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 22.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 283.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 264.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 392.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 231.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 90.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 199.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 425.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 423.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 57.8T + 3.00e5T^{2} \) |
| 71 | \( 1 + 201.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 380.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 927.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 152.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 552.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 254.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.951321138757405823707359934200, −7.51665158599101715901521396989, −6.98277087125697394971573735084, −5.93842824208432240960356874551, −5.63939067481538748025631084351, −4.67134767767415775598373836228, −3.74746830069189080624390049086, −3.27737134164980169201521121150, −2.04008437871280161250042292691, −1.01211692390796415754457620506,
1.01211692390796415754457620506, 2.04008437871280161250042292691, 3.27737134164980169201521121150, 3.74746830069189080624390049086, 4.67134767767415775598373836228, 5.63939067481538748025631084351, 5.93842824208432240960356874551, 6.98277087125697394971573735084, 7.51665158599101715901521396989, 8.951321138757405823707359934200
